In the case of low Reynolds number flows in such settings, elastic elongated filaments can be modelled using various simulation methods, for example: immersed boundary (IB) method [1]}, regularised singularity methods [2]}, bead-spring models [3]}, [4]}, [5]}, and discrete and continuous variants of slender-body theory (SBT) [6]}, [7]}. The reduced dimensionality of the filament offers a computational advantage, which has been used in variants of the IB technique to study the whirling instability of spinning filaments [8]} and hydrodynamic bundling of bacterial flagella [9]}. A combination of SBT and regularised Stokeslet method has also been formulated by [10]} and profitably applied e.g. to explain the motion of flagella in dinoflagellates [11]}.
See [12]} for a review of this approach. On the other hand, in methods which treat the filament as fundamentally one-dimensional, such as SBT, one faces problems when the mesh along the filament is too fine (small in comparison to the reduced length scale), even when smoothing the integral kernels, as discussed by [7]}.
Finally, any numerical scheme for the time evolution of elastic filaments must address the stiffness of the equations caused by presence of high-order spatial derivatives in the equations of motion responsible for bending rigidity. Due to the very high rate at which disturbances of small wavelength are damped, the issue of stiffness becomes even more pronounced with finer mesh sizes.
Practically all numerical works to date use an implicit integration scheme, while here we present a different approach.
